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1. If the sides of a pentagon, no two sides of which are parallel, be produced till they meet; show that the sum of all the angles at their points of intersection will be equal to two right angles.
2. Show that two chords which are equally distant from the centre of a circle are equal to each other; and of two chords, that which is nearer the centre is greater than the one more remote.
3. If through the angles of an isosceles triangle which has each of the angles at the base double of the third angle, and is inscribed in a circle, straight lines be drawn touching the circle ; show that an isosceles triangle will be formed which has each of the angles at the base one-third of the angle at the vertex.
4. A D B is a semicircle of which the centre is C; and A EC is another semicircle on the diameter AC; A T is a common tangent to the two semicircles at the point A. Show that if from any point F, in the circumference of the first, a straight line FC be drawn to C, the part FK, cut off by the second semicircle, is equal to the perpendicular F H to the tangent A T.
5. Show that the bisectors of the angles contained by the opposite sides (produced) of an inscribed quadrilateral intersect at right angles.
6. If a triangle A B C be formed by the intersection of three tangents to a circumference whose centre is 0, two of which, A M and AN, are fixed, while the third, BC, touches the circumference at a variable point P; show that the perimeter of the triangle A B C is constant, and equal to A M + AN, or 2 A M. Also show that the angle B O C is constant.
7. A B is any chord and A C is tangent to a circle at A, C D E a line cutting the circumference in D and E and parallel to AB; show that the triangle A C D is equiangular to the triangle E A B.
1. Draw two concentric circles, such that the chords of the outer circle which touch the inner may be equal to the diameter of the inner circle.
2. Given the base of a triangle, the vertical angle, and the length of the line drawn from the vertex to the middle point of the base : construct the triangle.
3. Given a side of a triangle, its vertical angle, and the radius of the circumscribing circle : construct the triangle.
4. Given the base, vertical angle, and the perpendicular from the extremity of the base to the opposite side : construct the 'triangle. . 5. Describe a circle cutting the sides of a given square, so that its circumference may be divided at the points of intersection into eight equal arcs.
6. Construct an angle of 60°, one of 30°, one of 120°, one of 150°, one of 45°, and one of 135o.
7. In a given triangle A B C, draw Q D E parallel to the base BC and meeting the sides of the triangle at D and E, so that D E shall be equal to D B + EC.
8. Given two perpendiculars, A B and C D, intersecting in 0, and a straight line intersecting these perpendiculars in E and F; to construct a square, one of whose angles shall coincide with one of the right angles at 0, and the vertex of the opposite angle of the square shall lie in EF. (Two solutions.)
9. In a given rhombus to inscribe a square.
10. If the base and vertical angle of a triangle be given ; find the locus of the vertex. .
11. If a ladder, whose foot rests on a horizontal plane and top against a vertical wall, slip down; find the locus of its middle point.
PROPORTIONAL LINES AND SIMILAR POLYGONS.
ON THE THEORY OF PROPORTION.
245. DEF. The Terms of a ratio are the quantities com
246. DEF. The Antecedent of a ratio is its first tern.
248. DEF. A Proportion is an expression of equality between two equal ratios.
A proportion may be expressed in any one of the following forms:
1. a : 6 ::6:d
Form 1 is read, a is to b as c is to d.
The Terms of a proportion are the four quantities compared.
The first and third terms in a proportion are the antecedents, the second and fourth terms are the consequents.
249. The Extremes in a proportion are the first and fourth terms.
250. The Means in a proportion are the second and third . terms.
251. DEF. In the proportion a :b::c:d; d is a Fourth Proportional to a, b, and c.
252. DEF. In the proportion a :b::b:c; c is a Third Proportional to a and b.
253. DEF. In the proportion a : 6::b:c; b is a Mean Proportional between a and c.
254. DEF. Four quantities are Reciprocally Proportional when the first is to the second as the reciprocal of the third is to the reciprocal of the fourth.
, 1 1 Thus
If we have two quantities a and b, and the reciprocals of these quantities - and ; these four quantities form a reciprocal proportion, the first being to the second as the reciprocal of the second is to the reciprocal of the first.
255. DEF. A proportion is taken by Alternation, when the means, or the extremes, are made to exchange places. Thus in the proportion
a ::::d, we have either
a :c ::6:d, or, d : 6 :: 0 : a. 256. DEF. A proportion is taken by Inversion, when the means and extremes are made to exchange places. Thus in the proportion
. a :b :: 0 :d, by inversion we have
b: a ::d : c.
257. Def. A proportion is taken by Composition, when the sum of the first and second is to the second as the sum of
the third and fourth is to the fourth ; or when the sum of the first and second is to the first as the sum of the third and fourth is to the third.
Thus if . a :b :: 0 :d,
258. Def. A proportion is taken by Division, when the difference of the first and second is to the second as the difference of the third and fourth is to the fourth ; or when the difference of the first and second is to the first as the difference of the third and fourth is to the third. Thus if
a :b :: :d,
259. In every numerical proportion the product of the extremes is equal to the product of the means.
. Let a : 6::c:d.